The Ideal Op-Amp lecture.pdf

Ideal

1. 2/9/2011 The Ideal Op-Amp lecture 1/13
Jim Stiles The Univ. of Kansas Dept. of EECS
The Ideal
Operational Amplifier

We begin by considering the equivalent circuit of an ideal op-amp:












Note that output voltage is defined with respect to ground potential, while the
input voltage is simply the potential difference between the plus (+) terminal
and the minus (-) terminal.

in
R


()
in
v
t
+ −

(
)
init

(
)
vo in
Avt

out
R

()
out
t
v
+ −

(
)
outit

+ −

+
_

2. 2/9/2011 The Ideal Op-Amp lecture 2/13
Jim Stiles The Univ. of Kansas Dept. of EECS
Very large and very small

Of course, we have three parameters in this circuit model: input resistance,
output resistance, and open circuit voltage gain.


Q:
So what are the ideal attributes (R
in
, R
out
, and A
vo
) of an operational
amplifier? In other words, what is the perfect op-amp?


A1: The input resistance of a perfect op-amp is infinitely large (i.e.,
in
R
=∞
).

A2: The output resistance of a perfect op-amp is zero (i.e.,
0
out
R
=
).

A3: The open-circuit voltage gain of a perfect op-amp is very large,
approaching infinity (
vo
A
≅
∞
)!

3. 2/9/2011 The Ideal Op-Amp lecture 3/13
Jim Stiles The Univ. of Kansas Dept. of EECS
The ideal op-amp model

Thus, the equivalent circuit model of an ideal op-amp is:











Here we have changed the notation of the open-circuit voltage gain.

The value
op
A
is used, where:
lim vo
op vo
A
AA
→∞



In other words, the gain value
op
A
is unfathomably large!

()
in
v
t
+ −

()
out
t
v
+ −

(
)
outit

+ −

+
_
()
op in
Avt
0
ini
=

4. 2/9/2011 The Ideal Op-Amp lecture 4/13
Jim Stiles The Univ. of Kansas Dept. of EECS
Ideal at all frequencies!

Note then:

1. Since the input resistance is infinite, the input current is zero—always!

2. Since the output resistance is zero, the output voltage is equal to the
open-circuit output voltage, even when the output load is not an open
circuit! I.E.,:

(
)
(
)
out op in
v
tAvt
=
? regardless of
outi
!


Q:
What about the
bandwidth
of this “ideal” op-amp; is the model
only
valid for
low-frequencies
?

A: Not for an ideal op-amp!

The bandwidth of an ideal op-amp is likewise infinite.

5. 2/9/2011 The Ideal Op-Amp lecture 5/13
Jim Stiles The Univ. of Kansas Dept. of EECS
It just seems so perfect


Q:
Wow! Unfathomably
high
voltage gain,
infinite
input resistance (impedance),
zero
output resistance (impedance), and:

(
)
(
)
out op in
v
tAvt
=


regardless
of the frequency spectrum
(
)
in
V
ω.

This sounds like the
perfect voltage amplifier
!

A: It is! That’s why we refer to it as the ideal op-amp.

+−

6. 2/9/2011 The Ideal Op-Amp lecture 6/13
Jim Stiles The Univ. of Kansas Dept. of EECS
Why the output but not the input?

Q:
So why isn’t the input voltage with respect to
ground
potential? Why is not
the minus (-) input terminal
connected to ground
?

A: Generally speaking, we find that two different voltages will be connected to
the two different input terminals:













()
in
v
t
+ −

()
out
t
v
+ −

(
)
outit

+ −

+
_
()
op in
Avt
+ _
(
)
2
v
t
(
)
1
v
t
+ _

7. 2/9/2011 The Ideal Op-Amp lecture 7/13
Jim Stiles The Univ. of Kansas Dept. of EECS
The input is a differential voltage

From KVL it is clear (right?) that the input voltage is:

(
)
(
)
(
)
21
in
v
tvtvt
=−


And so the output voltage is:

(
)
(
)
(
)
(
)
21
out op
v
tAvtvt
=−


Note that the input voltage is simply the difference between the two input
signals
2
v
and
1
v
.
We call this the differential input signal:

(
)
(
)
(
)
21
d
v
tvtvt
−

8. 2/9/2011 The Ideal Op-Amp lecture 8/13
Jim Stiles The Univ. of Kansas Dept. of EECS
It’s called a differential amplifier

Thus, we can likewise express the output as:

(
)
(
)
out op d
v
tAvt
=














Amplifiers of this type—where the input voltage is not defined with respect to
ground—are referred to as differential amplifiers, as they can amplify the
differential mode of two distinct signals (e.g.,
(
)
2
v
t
and
(
)
1
v
t
).

()
d
v
t
+ −

()
out
t
v
+ −

(
)
outit

+ −

+
_
(
)
op d
Avt
+ _
(
)
2
v
t
(
)
1
v
t
+ _

9. 2/9/2011 The Ideal Op-Amp lecture 9/13
Jim Stiles The Univ. of Kansas Dept. of EECS
An example

For example, say:

(
)
1

7.0 cos(10 ) 2.0 cos(5 )
v
tπtπt
=+


and:

(
)
2

7.0 cos(10 ) 5.0 cos(5 )
v
tπtπt
=+


the ideal op-amp output voltage is therefore:


(
)
(
)
(
)
(
)
21


3.0 cos(5 )
out op
op
v
tAvtvt
Aπt
=−
=



? What happened to
cos(10 )
πt
??

10. 2/9/2011 The Ideal Op-Amp lecture 10/13
Jim Stiles The Univ. of Kansas Dept. of EECS
The common mode disappears!

Note:

1. The difference between
(
)
2
v
t
and
(
)
1
v
t
is amplified.
2. The common signal (7 cos 10π
t
) is eliminated by the subtraction .

)

Difference amplifiers ideally have perfect common-mode rejection. That
is, the common signal between the two inputs has no effect on the output signal.

Of course, we can always connect a terminal to ground potential (i.e.,
(
)
1
0
v
t
=
),
thus making the input voltage a value with respect to ground:

()
in
v
t
+ −

()
out
t
v
+ −

(
)
outit

+ −

+
_
(
)
2
op
Avt
+ _
(
)
2
v
t

11. 2/9/2011 The Ideal Op-Amp lecture 11/13
Jim Stiles The Univ. of Kansas Dept. of EECS
The ideal op-amp; is it bogus?

















A: It is true that the output voltage will be very large— unless the differential
voltage is unfathomably small!


Q:
I scoff at your s
o
-called
“ideal”
o
p
-amp.


Although

and 0
in out
RR
=
∞=
are obviously correct, I
deem your assertion that A
op
should be unfathomably
large (approaching
∞
) to be a silly notion.

After all, a gigantic gain A
op
would mean that the output
voltage
21
()
out op
v
Av v
=
−
would likewise be unfathomably
large—the destructive implications are obvious.

12. 2/9/2011 The Ideal Op-Amp lecture 12/13
Jim Stiles The Univ. of Kansas Dept. of EECS
Definitely not bogus!

For example, what if the differential voltage is approximately (i.e., almost)
zero:
(
)
(
)()
21
0
d
v
tvtvt
≅⇒ ≅
?

In this case, the output voltage may not be very large at all!


Q:
Yes, but what is the
likelihood
that the two voltages
(
)
2
v
t
and
(
)
1
v
t
are
nearly the same? This seems
improbable
.

A: Op-amps are generally not implemented
by themselves!



Instead, they typically are but one
component of many in a feedback
amplifier.

+

-

v
in
(
t
)

ideal
C

R
1
v
1
v
2

i
2
(
t
)
v
out
(
t
)

i
1
(
t
)

13. 2/9/2011 The Ideal Op-Amp lecture 13/13
Jim Stiles The Univ. of Kansas Dept. of EECS
Get used to the virtual short!

In these applications, we will indeed find that
21
v
v
≅
—but we will also find that
this is a desirable condition!
The condition
21
v
v
≅
is know as a virtual short.




If this is not true, the output voltage of an
ideal op-amp will be unfathomably large.






As a result, the virtual short
21
v
v
≅
is almost always the case in useful op-amp
circuits.